Method and apparatus for constructing a polar code

ABSTRACT

A polar code construction method, apparatus, electronic device, and readable storage medium, applied to the field of wireless communication technology to reduce the complexity of polar code construction. The method comprises: calculating a weight spectrum corresponding to a polarized channel of a polar code with code length 2N based on the weight spectrum corresponding to the polarized channel with code length N, and MacWilliams identities; calculating, for each polarized channel, a union bound on the error probability of the polarized channel under the condition of additive white Gaussian noise based on the weight spectrum corresponding to the polarized channel and a union bound formula; determining the error probability threshold values based on the union bounds and a measurement method; sorting the error probability threshold values of all the polarized channels in ascending order, and selecting the polarized channels corresponding to the K smallest error probability threshold values.

The present application is a national stage application of PCTApplication No. PCT/CN2019/093040 which claims the priority to a ChinesePatent Application No. 201811494300.4, filed with the China NationalIntellectual Property Administration on Dec. 7, 2018 and entitled “APOLAR CODE CONSTRUCTION METHOD, APPARATUS, ELECTRONIC DEVICE, ANDREADABLE STORAGE MEDIUM”, each of which is incorporated herein byreference in its entirety.

TECHNICAL FIELD

The present application relates to the field of wireless communicationtechnology, and in particular to a method, apparatus, electronic device,and readable storage medium for constructing a polar code.

BACKGROUND

Additive white Gaussian noise (AWGN) is a common noise model in wirelesscommunication, where the amplitude of the noise obeys the normaldistribution, and the power spectral density of the noise is uniformlydistributed. A channel having AWGN is named as an AWGN channel, which isan ideal channel model usually used in a communication system. TheGaussian approximation (GA) is a simplified method of the densityevolution (DE) algorithm, which is used for estimating the reliabilitiesof the polarized channels in a polar code, that is, estimating errorprobabilities corresponding to the polarized channels. In the binaryadditive white Gaussian noise channel (BAWGNC), the GA algorithmapproximates the logarithm likelihood ratio (LLR) in the DE method witha set of Gaussian distributed random variables whose variance is twicethe mean. Since the Gaussian distributed probability density function(PDF) can be completely determined by the mean and variance, one canmaster the PDF of the LLR by tracking the variation of the mean underthe assumption of Gaussian approximation.

However, in the practical digital communication system, whenconstructing a polar code by means of the GA algorithm, it is requiredto construct it per signal-to-noise ratio, that is, when thesignal-to-noise ratio changes, it is required to recalculate the errorprobabilities of the polarized channels, which leads to highcomputational complexity and thus low practicality.

SUMMARY

In order to reduce the complexity of constructing a polar code andimprove the practicality, this application aims to provide a polar codeconstruction method, apparatus, electronic device, and readable storagemedium. The specific technical solutions are as follows.

An embodiment of the present application provides a method forconstructing a polar code, and the method comprises:

calculating a polar weight spectrum A_(2N) ^((l))(p) corresponding to anl^(th) polarized channel of a polar code with code length 2N based on apolar weight spectrum A_(N) ^((i))(j) corresponding to an i^(th)polarized channel of a polar code with code length N and a formula

$\left\{ {\begin{matrix}{{{A_{2N}^{(l)}(p)} = {A_{N}^{(i)}(j)}},{p = {2\; j}},} \\{{{A_{2N}^{(l)}(p)} = 0},{p\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{positive}\mspace{14mu}{odd}\mspace{14mu}{number}\mspace{14mu}{less}\mspace{14mu}{than}\mspace{14mu} 2\; N}}\end{matrix},} \right.$where i=1, 2, . . . , N, l=i+N, j=0, 1, 2, . . . , N, j denotes a weightof a codeword in the polar code with code length N, p denotes a weightof a codeword in the polar code with code length 2N, N=2^(x), x is apositive integer;

calculating a polar weight spectrum A_(2N) ^((m))(p) corresponding to anm^(th) polarized channel of the polar code with code length 2N, based onthe polar weight spectrum A_(2N) ^((l))(p) corresponding to the l^(th)polarized channel and the MacWilliams identities, where m=1, 2, . . . ,N;

calculating, for each polarized channel, a union bound on an errorprobability of the polarized channel under the condition of additivewhite Gaussian noise channel based on the polar weight spectrumcorresponding to the polarized channel of the polar code with codelength 2N and the union bound calculation formula;

determining the error probability threshold values of the polarizedchannels based on the union bounds of the polarized channels and apredetermined measurement method;

given the code length 2N and code rate K/2N, sorting the errorprobability threshold values of all the polarized channels in ascendingorder, and selecting polarized channels corresponding to the K smallesterror probability threshold values to transmit information bits, whileselecting the rest of polarized channels to transmit frozen bits; whereK denotes a length of information bits.

Optionally, determining the error probability threshold values of thepolarized channels based on the union bounds of the polarized channelsand a predetermined measurement method, comprises:

for each of the polarized channels, determining the union bound of thepolarized channel as the error probability threshold value of thepolarized channel; or

determining the logarithmic form of the union bound of the polarizedchannel as the error probability threshold value of the polarizedchannel; or

selecting the item corresponding to the minimum Hamming weight in thelogarithmic form of the union bound of the polarized channel as theerror probability threshold value of the polarized channel.

Optionally, calculating the polar weight spectrum A_(2N) ^((m))(p)corresponding to the m^(th) polarized channel of the polar code withcode length 2N, based on the polar weight spectrum A_(2N) ^((l))(p)corresponding to the l^(th) polarized channel and the MacWilliamsidentities, comprises:

calculating the weight spectrum S_(2N) ^((l))(p) corresponding to thel^(th) polarized channel of the polar code with code length 2N by usingthe formula

${{S_{2\; N}^{(l)}(p)} = {\sum\limits_{k = l}^{2N}{A_{2N}^{(k)}(p)}}};$

calculating the weight spectrum S_(2N) ^((m))(p) corresponding to them^(th) polarized channel of the polar code with code length 2N based onthe MacWilliams identities

${{\sum\limits_{p = 0}^{{2N} - v}{\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}{S_{2N}^{(m)}(p)}}} = {2^{{2N} - m + 1 - v}{\sum\limits_{p = 0}^{v}{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix}{S_{2N}^{({{2N} + 2 - m})}(p)}}}}},{{{where}\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}} = \frac{\left( {{2N} - p} \right)!}{{v!}{\left( {{2N} - p - v} \right)!}}},{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix} = \frac{\left( {{2N} - p} \right)!}{{\left( {{2N} - v} \right)!}{\left( {v - p} \right)!}}},{{{and}\mspace{14mu} v} = 0},1,\ldots\mspace{14mu},{{2N};}$

calculating the polar weight spectrum A_(2N) ^((m))(p) corresponding tothe m^(th) polarized channel of the polar code with code length 2N basedon A_(2N) ^((m))(p)=S_(2N) ^((m))(p)−S_(2N) ^((m+1))(p).

Optionally, calculating the union bound on the error probability of thepolarized channel under the condition of additive white Gaussian noisechannel based on the polar weight spectrum corresponding to thepolarized channel of the polar code with code length 2N and the unionbound calculation formula, comprises:

determining the union bound of the q^(th) polarized channel under thecondition of the additive white Gaussian noise channel as

${P_{UB}\left( W_{2N}^{(q)} \right)} = {\sum\limits_{p}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\;\frac{E_{s}}{N_{0}}} \right\}}}$by using the formula

${{P\left( W_{2N}^{(q)} \right)} \leq {\sum\limits_{p}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\;\frac{E_{s}}{N_{0}}} \right\}}}},$where q=1, 2, . . . , 2N, denotes the error probability of the q^(th)polarized channel, E_(s) denotes the average energy of the transmittedsignal, N₀ denotes the noise power spectral density, and E_(s)/N₀denotes the symbol signal-to-noise ratio (SNR).

An embodiment of the present application provides an apparatus forconstructing a polar code, the apparatus comprises:

a first module to calculate the polar weight spectrum, configured forcalculating a polar weight spectrum A_(2N) ^((l))(p) corresponding tothe l^(th) polarized channel of the polar code with code length 2N,based on the polar weight spectrum corresponding to the i^(th) polarizedchannel of the polar code with code length N and the formula

$\left\{ {\begin{matrix}{{{A_{2N}^{(l)}(p)} = {A_{N}^{(i)}(j)}},{p = {2j}},} \\{{{A_{2N}^{(l)}(p)} = 0},{p\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{positive}\mspace{14mu}{odd}\mspace{14mu}{number}\mspace{14mu}{less}\mspace{14mu}{than}\mspace{14mu} 2N}}\end{matrix},} \right.$where i=1, 2, . . . , N, l=i+N, j=0, 1, 2, . . . , N, j denotes theweight of a codeword in the polar code with code length N, p denotes theweight of a codeword in the polar code with code length 2N, N=2^(x), andx is a positive integer;

a second module to calculate the polar weight spectrum, configured forcalculating the polar weight spectrum A_(2N) ^((m))(p) corresponding tothe m^(th) polarized channel of the polar code with code length 2N,based on the polar weight spectrum A_(2N) ^((l))(p) corresponding to thel^(th) polarized channel and the MacWilliams identities, where m=1, 2, .. . , N;

a module to determine the union bound, configured for calculating, foreach polarized channel, the union bound on the error probability of thepolarized channel under the condition of additive white Gaussian noisechannel based on the polar weight spectrum corresponding to thepolarized channel of the polar code with code length 2N and the unionbound calculation formula;

a module to determine the error probability threshold value, configuredfor determining the error probability threshold values of the polarizedchannels based on the union bounds of the polarized channels and apredetermined measurement method;

a module to construct a polar code, configured for, given the codelength 2N and code rate K/2N, sorting the error probability thresholdvalues of all the polarized channels in ascending order, and selectingthe polarized channels corresponding to the K smallest error probabilitythreshold values to transmit information bits, while selecting the restof the polarized channels to transmit frozen bits, where K denotes thelength of information bits.

Optionally, the module to determine the error probability thresholdvalue comprises:

a first sub-module, configured for, for each of the polarized channels,determining the union bound of the polarized channel as the errorprobability threshold value of the polarized channel;

a second sub-module, configured for, for each of the polarized channels,determining the logarithmic form of the union bound of the polarizedchannel as the error probability threshold value of the polarizedchannel;

a third sub-module, configured for, for each of the polarized channels,selecting the item corresponding to the minimum Hamming weight in thelogarithmic form of the union bound of the polarized channel as theerror probability threshold value of the polarized channel.

Optionally, the second module to calculate the polar weight spectrum isspecifically configured for: calculating the weight spectrum S_(2N)^((l))(p) corresponding to the l^(th) polarized channel of the polarcode with code length 2N by using the formula

${{S_{2N}^{(l)}(p)} = {\sum\limits_{k = l}^{2N}{A_{2N}^{(h)}(p)}}};$

calculating the weight spectrum S_(2N) ^((m))(p) corresponding to them^(th) polarized channel of the polar code with code length 2N based onthe MacWilliams identities

${{\sum\limits_{p = 0}^{{2N} - v}{\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}{S_{2N}^{(m)}(p)}}} = {2^{{2N} - m + 1 - v}{\sum\limits_{p = 0}^{v}{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix}{S_{2N}^{({{2N} + 2 - m})}(p)}}}}},{{{where}\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}} = \frac{\left( {{2N} - p} \right)!}{{v!}{\left( {{2N} - p - v} \right)!}}},{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix} = \frac{\left( {{2N} - p} \right)!}{{\left( {{2N} - v} \right)!}{\left( {v - p} \right)!}}},{{{and}\mspace{14mu} v} = 0},1,\ldots\mspace{14mu},{{2N};}$

calculating the polar weight spectrum A_(2N) ^((m))(p) corresponding tothe m^(th) polarized channel of the polar code with code length 2N byusing A_(2N) ^((m))(p)=S_(2N) ^((m))(p)−S_(2N) ^((m+1))(p).

Optionally, the module to determine the union bound is specificallyconfigured for: determining the union bound of the q^(th) polarizedchannel under the condition of the additive white Gaussian noise channelas

${P_{UB}\left( W_{2N}^{(q)} \right)} = {\sum\limits_{p}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\;\frac{E_{z}}{N_{0}}} \right\}}}$by using the formula

${{P\left( W_{2N}^{(q)} \right)} \leq {\sum\limits_{p}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\;\frac{E_{s}}{N_{0}}} \right\}}}},$where q=1, 2, . . . , 2N, P(W_(2N) ^((q))) denotes the error probabilityof the q^(th) polarized channel, E_(s) denotes the average energy of thetransmitted signal, N₀ denotes the noise power spectral density, andE_(s)/N₀ denotes the symbol signal-to-noise ratio (SNR).

An embodiment of the present application provides an electronic device.The electronic device comprises: a processor, a communication interface,a memory and a communication bus, where the processor, the communicationinterface and the memory communicate with each other via thecommunication bus;

the memory is configured for storing a computer program;

the processor is configured for implementing any above steps of themethod for constructing a polar code when executing the computer programstored in the memory.

An embodiment of the present application provides a computer-readablestorage medium. The computer-readable storage medium stores a computerprogram therein, and the computer program is executed by a processor toimplement steps of the method for constructing a polar code.

In the polar code construction method, apparatus, electronic device, andreadable storage medium provided by embodiments of the presentapplication: the polar weight spectrum corresponding to each polarizedchannel of the polar code with code length 2N is calculated, based onthe polar weight spectrum corresponding to each polarized channel of thepolar code with code length N and the MacWilliams identities; the unionbound of each polarized channel under the condition of the additivewhite Gaussian noise channel is calculated based on the polar weightspectrum corresponding to each polarized channel of the polar code withcode length 2N and the union bound calculation formula; the errorprobability threshold value of each polarized channel is determinedbased on the union bound and a predetermined measurement method; giventhe code length 2N and code rate K/2N, the error probability thresholdvalues are sorted in ascending order, then the polarized channelscorresponding to the K smallest error probability threshold values areselected to transmit information bits, and the rest of the polarizedchannels are selected to transmit frozen bits. The construction methodof polar codes provided by the embodiments of this application can beSNR-independent, which reduces the complexity for constructing a polarcode, and has a good practical prospect for the polar coded transmissionsystem.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to describe the technical solutions of the embodiments of thisapplication and that of the prior art more clearly, drawings used in theembodiments and the prior art will be briefly described below.Obviously, the drawings described below only correspond to someembodiments of this application, and the technicians in this field canalso obtain other drawings based on these drawings without any creativeefforts.

FIG. 1 shows a flowchart of the method for constructing a polar codeaccording to an embodiment of the present application.

FIG. 2 compares the block error rate performances of polar codesconstructed by the Gaussian approximation algorithm and the methodproposed by the embodiment of this application.

FIG. 3 shows a structural diagram of an apparatus for constructing apolar code according to an embodiment of this application.

FIG. 4 shows a structural diagram of an electronic device according toan embodiment of this application.

DETAILED DESCRIPTION

In order to make the objectives, technical solutions and advantages ofthe present application clearer and more understandable, the presentapplication will be further described in detail below with reference tothe drawings and embodiments. Obviously, the described embodiments areonly part, and not all, of the embodiments of the present application.All other embodiments obtained based on the embodiments of thisapplication by the technicians in this field without any creativeefforts fall into the scope of protection defined by the presentapplication.

In order to solve the problem that the computational complexity ofconstructing a polar code is too high for practical application, theembodiments of the present application provide a polar code constructionmethod, an apparatus, an electronic device and a readable storage mediumto reduce the complexity of constructing a polar code.

Firstly, the method for constructing a polar code provided by anembodiment of the present application is described in detail below.

Referring to FIG. 1, FIG. 1 shows a flowchart of the method forconstructing a polar code according to an embodiment of the presentapplication. The method comprises the following steps.

S101, calculate the polar weight spectrum A_(2N) ^((l))(p) correspondingto the l^(th) polarized channel of the polar code with code length 2N,based on the polar weight spectrum A_(2N) ^((i))(j) corresponding to thei^(th) polarized channel of the polar code with code length N and usingthe formula

$\left\{ {\begin{matrix}{{{A_{2N}^{(l)}(p)} = {A_{N}^{(i)}(j)}},{p = {2j}},} \\{{{A_{2N}^{(l)}(p)} = 0},{p\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{positive}\mspace{14mu}{odd}\mspace{14mu}{number}\mspace{14mu}{less}\mspace{14mu}{than}\mspace{14mu} 2N}}\end{matrix},} \right.$where i=1, 2, . . . , N, l=i+N, j=0, 1, 2, . . . , N, j denotes theweight of a codeword in the polar code with code length N, p denotes theweight of a codeword in the polar code with code length 2N, N=2^(x), andx is a positive integer.

In the embodiment of the present application, if the generator matrixfor the polar code with code length N is represented as F_(N), and thegenerator matrix for the polar code with code length 2N is representedas F_(2N), then there is a recursive relationship:

$F_{2N} = {\begin{bmatrix}F_{N} & 0 \\F_{N} & F_{N}\end{bmatrix}.}$For example, if there is

${F_{2} = \begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}},$then for a polar code with N=4, there is

$F_{4} = {\begin{bmatrix}1 & 0 & 0 & 0 \\1 & 1 & 0 & 0 \\1 & 0 & 1 & 0 \\1 & 1 & 1 & 1\end{bmatrix}.}$

N polarized channels will be obtained after the channel polarizationtransformation is performed on the polar code with code length N, andthe index of the polarized channel is denoted by i.

A codeword vector of the polar code is obtained by e=uF_(N), where u isthe 1×N source information vector.

For the polar code with N=4,

${F_{4} = \begin{bmatrix}1 & 0 & 0 & 0 \\1 & 1 & 0 & 0 \\1 & 0 & 1 & 0 \\1 & 1 & 1 & 1\end{bmatrix}},$and four polarized channels will be obtained after the channelpolarization transformation. Based on the relationship between polarizedchannels and linear block codes, the linear block code corresponding tothe polarized channel with i=4 only uses the 4^(th) row of the generatormatrix F₄. Then the first three bits of the source information vector uare fixed as 0, and only the 4^(th) bit can be chosen randomly as 0or 1. Therefore, the codewords corresponding to the polarized channelwith i=4 can be obtained as the following Table 1.

TABLE 1 u c = uF₄ j [0 0 0 0] [0 0 0 0] 0 [0 0 0 1] [1 1 1 1] 4

As shown in Table 1, for the polarized channel with i=4, there are twocodewords. It can be observed from the third column of Table 1, theHamming weights of the two codewords are 0 and 4, respectively.

Let the weight spectrum S_(N) ^((i))(j) denote the number of codewordswith Hamming weight j in the linear block code corresponding to thepolarized channel with index i. Thus, the weight spectrum of thepolarized channel with i=4 can be obtained as the Table 2 below.

TABLE 2 i j S₄ ^((i)) (j) 4 0 1 4 4 1

Based on the relationship between the polarized channels and the linearblock codes, the linear block code corresponding to the polarizedchannel with i=3 only uses the 3^(rd) and 4^(th) rows of the generatormatrix F₄. Then the first two bits of the source information vector uare fixed as 0, and the last two bits can be chosen randomly as 0 or 1.Therefore, the codewords corresponding to the polarized channel with i=3can be obtained as the following Table 3.

TABLE 3 u c = uF₄ j [0 0 0 0] [0 0 0 0] 0 [0 0 0 1] [1 1 1 1] 4 [0 0 10] [1 0 1 0] 2 [0 0 1 1] [0 1 0 1] 2

As shown in Table 3, for the polarized channel with i=3, there are fourcodewords, including one with weight 0, one with weight 4, and two withweight 2. Thus, the weight spectrum of the polarized channel with i=3can be obtained as the Table 4 below.

TABLE 4 i j S₄ ^((i)) (j) 3 0 1 3 2 2 3 4 1

Similarly, when i=2, the linear block code corresponding to thepolarized channel with i=2 only uses the last three rows of thegenerator matrix F₄, then the 1^(st) bit of the source informationvector u is fixed as 0, and the last three bits can be chosen randomlyas 0 or 1. Therefore, the codewords corresponding to the polarizedchannel with i=2 can be obtained as the following Table 5.

TABLE 5 u c = uF₄ j [0 0 0 0] [0 0 0 0] 0 [0 0 0 1] [1 1 1 1] 4 [0 0 10] [1 0 1 0] 2 [0 0 1 1] [0 1 0 1] 2 [0 1 0 0] [1 1 0 0] 2 [0 1 0 1] [00 1 1] 2 [0 1 1 0] [0 1 1 0] 2 [0 1 1 1] [1 0 0 1] 2

As shown in Table 5, for the polarized channel with i=2, there are 8codewords, including one with weight 0, one with weight 4, and six withweight 2. Thus, the weight spectrum of the polarized channel with i=2can be obtained as the Table 6 below.

TABLE 6 i j S₄ ^((i)) (j) 2 0 1 2 2 6 2 4 1

Similarly, when i=1, the linear block code corresponding to thepolarized channel with i=1 uses the generator matrix F₄, then all thebits of the source information vector u can be chosen randomly as 0or 1. It is straightforward to know that there are 16 codewords for thepolarized channel with i=1 and the corresponding weight spectrum can beshown in Table 7.

TABLE 7 i j S₄ ^((i)) (j) 1 0 1 1 1 4 1 2 6 1 3 4 1 4 1

Combining the above weight spectrums corresponding to all the polarizedchannels, the complete weight spectrum of the polar code with N=4 isobtained as shown in Table 8.

TABLE 8 i j S₄ ^((i)) (j) 1 0 1 1 1 4 1 2 6 1 3 4 1 4 1 2 0 1 2 2 6 2 41 3 0 1 3 2 2 3 4 1 4 0 1 4 4 1

Comparing the Table 1 and Table 3, it can be observed that the 2codewords corresponding to the polarized channel with i=4 are includedin the 4 codewords corresponding to the polarized channel with i=3;similarly, the 8 codewords corresponding to i=2 are included in the 16codewords corresponding to i=1. Therefore, a concept named polar weightspectrum is proposed in the embodiment of the present application. Thepolar weight spectrum corresponding to each polarized channel does notcomprise the codewords corresponding to the next polarized channel.

Therefore, the polar weight spectrum can be calculated by the weightspectrum A_(N) ^((i))(j)=S_(N) ^((i))(j)−S_(N) ^((i+1))(j), for i=1, 2,. . . , N−1, and A_(N) ^((i))(0)=0, for i=1, 2, . . . , N. The polarweight spectrum of the polar code with N=4 is provided in the followingTable 9.

TABLE 9 i j A₄ ^((i)) (j) 1 1 4 1 3 4 2 2 4 3 2 2 4 4 1

By using the similar method as described above, the polar weightspectrum of the polar code with code length N=8, 16, 32, . . . also canbe obtained.

Given the polar weight spectrum of the polar code with code length N,for the polar code with code length 2N, if the index of the polarizedchannel satisfies N+1≤l≤2N, then based on the structure of the matrixF_(2N), the polarized channel is located in the lower half part of thematrix F_(2N), which is composed by partial rows of two identical F_(N)matrixes repeatedly. Therefore, the l^(th) polarized channel of thepolar code with code length 2N corresponds to the i^(th) polarizedchannel of the polar code with code length N, where N+1≤l≤2N, i=l−N. Therelation exists between their corresponding polar weight spectrums isthat: A_(2N) ^((l))(p)=A_(N) ^((i))(j) and p=2j. When the code weight pis a positive odd number, the corresponding numbers of codewords are 0.

The polar weight spectrum shown in Table 9 can be equivalentlytransformed as that in Table 10 below.

TABLE 10 j polar weight spectrum i 0 1 2 3 4 4 0 0 0 0 1 3 0 0 2 0 0 2 00 4 0 0 1 0 4 0 4 0

The polar weight spectrums of the lower half polarized channels (5≤l≤8)of a polar code with code length 8 can be directly obtained based on thepolar weight spectrum of the polar code with code length 4 and a formula

$\left\{ {\begin{matrix}{{{A_{2N}^{(l)}(p)} = {A_{N}^{(i)}(j)}},{p = {2j}},} \\{{{A_{2N}^{(l)}(p)} = 0},{p\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{positive}\mspace{14mu}{odd}\mspace{14mu}{number}\mspace{14mu}{less}\mspace{14mu}{than}\mspace{14mu} 2N}}\end{matrix}.} \right.$The method for calculating the polar weight spectrums of the upper halfpolarized channels (1≤l≤4) of the polar code with code length 8 will bedescribed in the following step. The polar weight spectrums of the lowerhalf polarized channels (5≤l≤8) of the polar code with code length 8 areshown in the following Table 11.

TABLE 11 p polar weight spectrum l 0 1 2 3 4 5 6 7 8 8 0 0 0 0 0 0 0 0 17 0 0 0 0 2 0 0 0 0 6 0 0 0 0 4 0 0 0 0 5 0 0 4 0 0 0 4 0 0

It can be seen from Table 11 that for 5≤l≤8, the number of the codewordswith odd weight is 0 while the number of the codewords with even weightis equal to the corresponding item in Table 10.

S102, calculate the polar weight spectrum A_(2N) ^((m))(p) correspondingto the m^(th) polarized channel of the polar code with code length 2N,based on the polar weight spectrum A_(2N) ^((l))(p) corresponding to thel^(th) polarized channel and the MacWilliams identities, where m=1, 2, .. . , N;

Specifically, for m=1, 2, . . . , N, based on the structure of thematrix F_(2N), the index of the polarized channel m is corresponding tothe upper half of the matrix F_(2N), which is composed by partial rowvectors of the F_(N) matrix. For the polar code with code length 2N, thedual code of the (2N, 2N−m+1) linear block code corresponding to them^(th) polarized channel is a (2N, m−1) linear block code. Hence, it canbe derived that the dual polarized channel of the m^(th) polarizedchannel is the one with index 2N−(m−1)+1, that is, 2N−m+2. Therefore,for the polarized channels with m=2, 3, . . . , N, their correspondingdual polarized channels can be found in the lower half part of thepolarized channels described in S101. Meanwhile, the dual code of the(2N,2N−m+1) linear block code corresponding to the m^(th) polarizedchannel also can be found in S101.

For example, the duality relationships between the polarized channels ofthe polar code with code length 8 are shown in the following Table 12.

TABLE 12 channel index index of the dual channel 8 2 7 3 6 4 5 5

It can be found from Table 12 that the polarized channel with m=1 doesnot have a dual polarized channel, therefore, its polar weight spectrumwill be calculated later.

Given the code length 2N, the weight spectrums of the polarized channelswith index N+1, N+2, . . . , 2N can be determined based on thecorresponding polar weight spectrums. Then, the weight spectrums of thepolarized channels with index 1, 2, . . . , N can be calculated via theMacWilliams identities because the weight spectrums of a pair of dualcodes satisfy the MacWilliams identities. Finally, the polar weightspectrums of the polarized channels with index 1, 2, . . . , N can becalculated by using the weight spectrums.

S103, calculate, for each polarized channel, the union bound on theerror probability of the polarized channel under the condition ofadditive white Gaussian noise channel based on the polar weight spectrumcorresponding to the polarized channel of the polar code with codelength 2N and the union bound calculation formula;

Under the condition of additive white Gaussian noise channel, an upperbound on the error probability of the polarized channel can becalculated based on the polar weight spectrum, which is named as theunion bound. The union bound can be used to characterize the reliabilityof the polarized channel, which means the smaller the union bound, thehigher the reliability of the polarized channel.

S104, determine the error probability threshold values of the polarizedchannels based on the union bounds of the polarized channels and apredetermined measurement method.

In the embodiments of the present application, after the union bound ofeach polarized channel is obtained via S103, the error probabilitythreshold value of the polarized channel can be obtained by using thepredetermined measurement method. In this way, the error probabilitythreshold value of the polarized channel can be set according to thepractical requirement.

S105, given the code length 2N and code rate K/2N, sort the errorprobability threshold values of all the polarized channels in ascendingorder, and select the polarized channels corresponding to the K smallesterror probability threshold values to transmit information bits, whileselect the rest of the polarized channels to transmit frozen bits, whereK denotes the length of information bits.

Since the smaller the error probability threshold value is, the higherthe reliability of the polarized channel is, when constructing the polarcode with code length 2N and code rate K/2N, the polarized channelscorresponding to the K smallest error probability threshold values arechosen to transmit information bits while the others are selected totransmit frozen bits.

For example, given the code length 1024 and code rate 0.5, i.e., thelength of information bits is 512, the error probability thresholdvalues corresponding to all of 1024 polarized channels are calculatedfirstly when constructing the polar code. Then, all of the errorprobability threshold values are sorted in ascending order and thesmaller threshold value means the corresponding polarized channel ismore reliable. Therefore, when constructing a polar code, the polarizedchannels corresponding to the 512 smallest error probability thresholdvalues are chosen to transmit information bits while the others areselected to transmit frozen bits.

In the method for constructing a polar code provided by embodiments ofthis application: the polar weight spectrum corresponding to eachpolarized channel of the polar code with code length 2N is calculated,based on the polar weight spectrum corresponding to each polarizedchannel of the polar code with code length N and the MacWilliamsidentities; the union bound of each polarized channel under thecondition of the additive white Gaussian noise channel is calculatedbased on the polar weight spectrum corresponding to each polarizedchannel of the polar code with code length 2N and the union boundcalculation formula; the error probability threshold value of eachpolarized channel is determined based on the union bound and apredetermined measurement method; given the code length 2N and code rateK/2N, the error probability threshold values are sorted in ascendingorder, and the polarized channels corresponding to the K smallest errorprobability threshold values are selected to transmit information bits,while the rest of the polarized channels are selected to transmit frozenbits. The construction method of polar codes provided by the embodimentsof this application can be SNR-independent, which reduces the complexityfor constructing a polar code, and has a good practical prospect for thepolar coded transmission system.

An implementation of the present application, the embodiment S102 inFIG. 1 comprises the following steps.

Firstly, calculate the weight spectrum S_(2N) ^((l))(p) corresponding tothe l^(th) polarized channel of the polar code with code length 2N byusing

${S_{2N}^{(l)}(p)} = {\sum\limits_{h = l}^{2N}{{A_{2N}^{(h)}(p)}.}}$

Specifically, for the polar code with code length 2N, the polar weightspectrums of the polarized channels with index from N+1 to 2N have beencalculated in S101. In order to calculate the polar weight spectrums ofthe polarized channels with index from 1 to N by using the MacWilliamsidentities, the weight spectrums of the polarized channels with indexfrom N+1 to 2N, i.e., S_(2N) ^((l))(p) are required to be calculatedfirstly.

Specifically, the linear block code is usually represented as (2N,K),where 2N denotes the code length, K denotes the length of informationbits, and the code rate is denoted by R=K/2N. For the l^(th) polarizedchannel, the code rate of its corresponding linear block code is

${R = \frac{{2\; N} - l + 1}{2\; N}},$that is, the corresponding linear block code is a (2N,2N−l+1) code. Let{S_(2N) ^(l)(p)} denote the weight spectrum of the l^(th) polarizedchannel, where p (0≤p≤2N) is the Hamming weight of codeword and S_(2N)^((i))(p) is the number of codewords of weight p for the (2N,2N−l+1)linear block code. Similarly, let {A_(2N) ^(l)(p)} denote the polarweight spectrum, where A_(2N) ^(l)(p) is the number of codewords ofweight p for a subset of the (2N,2N−l+1) linear block code. The weightspectrum S_(2N) ^((l))(p (p≠0) of the l^(th) polarized channel isobtained by accumulating the polar weight spectrums of the polarizedchannels with index from l to 2N, and there is always one all-zerocodeword corresponds to each polarized channel, that is,

${{S_{2\; N}^{(l)}(p)} = {\sum\limits_{t = l}^{2\; N}\;{A_{n}^{(t)}(p)}}},{p = 1},2,\ldots\mspace{14mu},{2\; N},{{S_{2\; N}^{(l)}(p)} = 1},{p = 0.}$

For example, the polar weight spectrum of the polar code with codelength 4 is shown in Table 10 and the corresponding weight spectrum isshown in Table 13.

TABLE 13 j weight spectrum i 0 1 2 3 4 4 1 0 0 0 1 3 1 0 2 0 1 2 1 0 6 01 1 1 4 6 4 1

Secondly, calculate the weight spectrum S_(2N) ^((m))(p) correspondingto the m^(th) polarized channel of the polar code with code length 2Nbased on the MacWilliams identities

${{\sum\limits_{p = 0}^{{2\; N} - v}\;{\begin{pmatrix}{{2\; N} - p} \\v\end{pmatrix}{S_{2\; N}^{(m)}(p)}}} = {2^{{2\; N} - m + 1 - v}{\sum\limits_{p = 0}^{v}{\begin{pmatrix}{{2\; N} - p} \\{{2\; N} - v}\end{pmatrix}{S_{2\; N}^{({{2\; N} + 2 - m})}(p)}}}}},{{{where}\begin{pmatrix}{{2\; N} - p} \\v\end{pmatrix}} = \frac{\left( {{2\; N} - p} \right)!}{{v!}{\left( {{2\; N} - p - v} \right)!}}},{\begin{pmatrix}{{2\; N} - p} \\{{2\; N} - v}\end{pmatrix} = \frac{\left( {{2\; N} - p} \right)!}{{\left( {{2\; N} - v} \right)!}{\left( {v - p} \right)!}}},{{{and}\mspace{14mu} v} = 0},1,\ldots\mspace{14mu},{2\;{N.}}$

In this step, for each of the polarized channels with index m=2, 3, . .. , N, the code weight p=0, 1, 2, . . . , 2N, that is, there are 2N+1unknowns. Meanwhile, there are 2N+1 equations in total since v=0, 1, . .. , 2N and an equation can be obtained once the value of v isdetermined. In each of the equations, the range of p in the equationwill be determined after v is determined, where the range of p is {0, 1,. . . , 2N-v} for the left side of the equation and that is {0, 1, . . ., v} for the right side of the equation. It can be known that the numberof the unknowns is equal to the number of the equations, thus, for eachof polarized channels, its corresponding equation group has the uniquesolution.

Further, the above equation group can be solved iteratively according tothe triangular characteristics of the unknown coefficients to obtain theweight spectrum S_(2N) ^((m))(p).

Firstly, let v=2N, then an equation can be derived as

${\begin{pmatrix}{2\; N} \\{2\; N}\end{pmatrix}{S_{2\; N}^{(m)}(0)}} = {2^{{- m} + 1}{\sum\limits_{p = 0}^{2\; N}\;{\begin{pmatrix}{{2\; N} - p} \\0\end{pmatrix}{{S_{2\; N}^{({{2\; N} + 2 - m})}(p)}.}}}}$It can be found that there is only one unknown at the left side of theequation, and the right side of the equation is a known quantity becausethe weight spectrums S_(2N) ^(2N+2−m)(p), m=2, 3, . . . , N, p=0, 1, . .. , 2N have been calculated. Therefore, the unknown of the equation canbe directly calculated by

${S_{2\; N}^{(m)}(0)} = {2^{{- m} + 1}{\sum\limits_{p = 0}^{2\; N}{{S_{2\; N}^{({{2\; N} + 2 - m})}(p)}.}}}$

Then, let v=2N−1, then an equation can be derived as

${\sum\limits_{p = 0}^{1}{\begin{pmatrix}{{2\; N} - p} \\{{2\; N} - 1}\end{pmatrix}{S_{2\; N}^{(m)}(p)}}} = {2^{{- m} + 2}{\sum\limits_{p = 0}^{{2\; N} - 1}{\begin{pmatrix}{{2\; N} - p} \\1\end{pmatrix}{{S_{2\; N}^{({{2\; N} + 2 - m})}(p)}.}}}}$At this time, there are two unknowns S_(2N) ^((m))(0) and S_(2N)^((m))(1) at the left side of the equation, where S_(2N) ^((m))(0) isalready obtained from the previous step, therefore, the unknown S_(2N)^((m))(1) can be calculated by

${S_{2\; N}^{(m)}(1)} = {{2^{{- m} + 2}{\sum\limits_{p = 0}^{{2\; N} - 1}{\left( {{2\; N} - p} \right){S_{2\; N}^{({{2\; N} + 2 - m})}(p)}}}} - {2\; N \times {{S_{2\; N}^{(m)}(0)}.}}}$

In the same way, the values of S_(2N) ^((m))(2), S_(2N) ^((m))(3), . . ., S_(2N) ^((m))(2N) can be obtained in turn. In practical applications,except for the first one, the number of codewords with positive oddweight corresponding to the polarized channels is zero, i.e., S_(2N)^((m))(p)=0, where m=2, 3, . . . , N and p=1, 3, . . . , 2N−1.Therefore, the computational complexity of solving the equations can bereduced by half.

Finally, calculate the polar weight spectrum A_(2N) ^((m))(p)corresponding to the m^(th) polarized channel of the polar code withcode length 2N by using A_(2N) ^((m))(p)=S_(2N) ^((m))(p)−S_(2N)^((m+1))(p), m=2, 3, . . . , N, p=0, 1, . . . , 2N.

In the embodiment of this application, the polar weight spectrum of eachpolarized channel should be calculated as above because it will be usedfor calculating the union bound on the error probability of thepolarized channel. Therefore, the polar weight spectrums of polarizedchannels (m=2, 3, . . . , N) is calculated based on the formulaA _(2N) ^((m))(p)=S _(2N) ^((m))(p)−S _(2N) ^((m+1))(p).

So far, only the polar weight spectrum of the first polarized channel(m=1) is unknown. Considering the characteristic of the polar code, thetotal number of codewords of weight p satisfies the binomialdistribution, that is, the total is

$\begin{pmatrix}{2N} \\p\end{pmatrix}.$Therefore, the polar weight spectrum of the first polarized channel canbe obtained by subtracting those of the rest polarized channels from thetotal, that is,

${{A_{2\; N}^{(1)}(p)} = {\begin{pmatrix}{2N} \\p\end{pmatrix} - {S_{2\; N}^{(2)}(p)}}},{\begin{pmatrix}{2N} \\p\end{pmatrix} = {\frac{\left( {2\; N} \right)!}{{p!}{\left( {{2\; N} - p} \right)!}}.}}$

In the above step, to calculate the weight spectrum of the polar codewith code length 2N, it is only required to solve N−1 groups of linearequations with 2N+1 unknowns so that the complexity of solving eachlinear equation group is of O(N²), then the total complexity is O(N³),which is bearable for the medium code length. Theoretically, the methodcan be extended to any code length. The polar weight spectrum of thepolar code with code length 8 obtained above is shown in Table 14 below.

TABLE 14 polar weight spectrum l 0 1 2 3 4 5 6 7 8 8 0 0 0 0 0 0 0 0 1 70 0 0 0 2 0 0 0 0 6 0 0 0 0 4 0 0 0 0 5 0 0 4 0 0 0 4 0 0 4 0 0 0 0 16 00 0 0 3 0 0 8 0 16 0 8 0 0 2 0 0 16 0 32 0 16 0 0 1 0 8 0 56 0 56 0 8 0

In an implementation of the present application, the embodiment S103 inFIG. 1, the union bound of the q^(th) polarized channel under thecondition of additive white Gaussian noise channel can be determined as

${P_{UB}\left( W_{2\; N}^{(q)} \right)} = {\sum\limits_{p}\;{{A_{2\; N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}$by using the formula

${{P\left( W_{2\; N}^{(q)} \right)} \leq {\sum\limits_{p}\;{{A_{2\; N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}},$where q=1, 2, . . . , 2N, P(W_(2N) ^((q))) denotes the error probabilityof the q^(th) polarized channel, E_(s) denotes the average energy of thetransmitted signal, N₀ denotes the noise power spectral density, andE_(s)/N₀ denotes the symbol signal-to-noise ratio (SNR).

Accordingly, in S104, by using the Jacobian transformation on thelogarithmic form of the union bound, the first universal constructionmetric can be obtained as

${UBW}_{2\; N}^{(q)} = {\max\limits_{p}{\left\{ {{\ln\left\lbrack {A_{2\; N}^{(q)}(p)} \right\rbrack} - {p\frac{E_{s}}{N_{0}}}} \right\}.}}$Let L_(2N) ^((q))(p)=ln[A_(2N) ^((q))(p)] denote the logarithmic form ofthe polar weight spectrum, where ln(·) denotes the natural logarithmwith base e=2.71828 . . . , then the first universal construction metriccan be further expressed as

${{UBW_{2N}^{(q)}} = {\max\limits_{p}\left\{ {{L_{2N}^{(q)}(p)} - {p\frac{E_{s}}{N_{0}}}} \right\}}}.$The first universal construction metric can be transformed into asimplified SNR-independent universal construction metric by selecting anoptimized value of E_(s)/N₀, which can be determined according to thecode length and code rate in practical applications.

Under the condition of high signal-to-noise ratio, the reliability ofeach polarized channel is mainly determined by the polar weight spectrumcorresponding to the minimum Hamming weight. The Hamming weight of abinary codeword is equal to the number of 1s in the codeword. Therefore,the first universal construction metric given above can be simplified asthe second universal construction metric

${UBWW}_{2\; N}^{(q)} = {{\ln\left\lbrack {A_{2\; N}^{(q)}\left( p_{\min} \right)} \right\rbrack} - {p_{\min}{\frac{E_{s}}{N_{0}}.}}}$Let L_(2N) ^((q))(p_(min))=ln[A_(2N) ^((q))(p_(min))] denote thelogarithmic form of the polar weight spectrum corresponding to theminimum Hamming weight, then the second universal construction metriccan be further expressed as

${UBWW}_{2N}^{(q)} = {{L_{2N}^{(q)}\left( p_{\min} \right)} - {p_{\min}{\frac{E_{s}}{N_{0}}.}}}$Similarly, the value of the signal-to-noise ratio E_(s)/N₀ can be fixedto obtain a simplified universal construction metric that is independentof E_(s)/N₀, and the value of E_(s)/N₀ can be determined according tothe code length and code rate in practical applications.

An implementation of the present application, the embodiment S104 inFIG. 1 comprises:

for each of the polarized channels, determining the union bound of thepolarized channel as the error probability threshold value of thepolarized channel; or

determining the logarithmic form of the union bound of the polarizedchannel as the error probability threshold value of the polarizedchannel; or

selecting the item corresponding to the minimum Hamming weight in thelogarithmic form of the union bound of the polarized channel as theerror probability threshold value of the polarized channel.

In the embodiment of the present application, the union bound of thepolarized channel can be directly determined as the error probabilitythreshold value of the polarized channel. In addition, the firstuniversal construction metric derived based on the logarithmic form ofthe union bound also can be regarded as the error probability thresholdvalue. Finally, since the reliability of each polarized channel ismainly determined by the polar weight spectrum corresponding to theminimum Hamming weight in the case of high signal-to-noise ratio, thesecond universal construction metric also can be regarded as the errorprobability threshold value and obtained based on the first universalconstruction metric by only considering the logarithmic form of thepolar weight spectrum corresponding to the minimum Hamming weight. Ofcourse, there may be a variety of methods for determining the errorprobability threshold value; all the methods for determining the errorprobability threshold value based on the union bound belong to the scopeof protection of the present application.

For example, given the code length N=1024 and code rate 0.5, FIG. 2compares the block error rate performances of polar codes constructed bythe Gaussian approximation algorithm and the two universal constructionmetrics proposed by the embodiment of this application. The successivecancellation (SC) decoding and the successive cancellation list (SCL)decoding with list size 16 are respectively used to decode the polarcodes. The vertical axis denotes the block error rate of polar codeswhile the horizontal axis denotes the signal-to-noise ratio. The valueof E_(s)/N₀ in the first universal construction metric is fixed as 4 dBand that in the second universal construction metric is set to 3.3 dB.

It can be observed from FIG. 2 that the polar codes constructed by theuniversal construction metrics proposed by this application can achieveapproximate or even better performances than those constructed by theGaussian approximation algorithm. Besides, the two universalconstruction metrics proposed by this application are independent ofSNR, which leads to a much lower complexity of constructing polar codes.

According to the embodiment of the above method, an apparatus forconstructing a polar code is provided in the embodiment of thisapplication, whose structural diagram is shown in FIG. 3. The apparatuscomprises:

a first module to calculate the polar weight spectrum 301, configuredfor) calculating a polar weight spectrum A_(2N) ^((l))(p) correspondingto an l^(th) polarized channel of a polar code with code length 2N basedon a polar weight spectrum corresponding to an l^(th) polarized channelof a polar code with code length N and a formula

$\left\{ {\begin{matrix}{{{A_{2N}^{(l)}(p)} = {A_{N}^{(i)}(j)}},{p = {2j}},} \\{{{A_{2N}^{(l)}(p)} = 0},{p\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{positive}\mspace{14mu}{odd}\mspace{14mu}{number}\mspace{14mu}{less}\mspace{14mu}{then}\mspace{14mu} 2N}}\end{matrix},} \right.$where i=1, 2, . . . , N, l=i+N, j=0, 1, 2, . . . , N, j denotes a weightof a codeword in the polar code with code length N, p denotes a weightof a codeword in the polar code with code length 2N, N=2^(x), x is apositive integer;

a second module to calculate the polar weight spectrum 302, configuredfor calculating a polar weight spectrum A_(2N) ^((m))(p) correspondingto an m^(th) polarized channel of the polar code with code length 2N,based on the polar weight spectrum A_(2N) ^((l))(p) corresponding to thel^(th) polarized channel and the MacWilliams identities, where m=1, 2, .. . , N;

a module to determine the union bound 303, configured for calculating,for each polarized channel, a union bound on an error probability of thepolarized channel under the condition of additive white Gaussian noisechannel based on the polar weight spectrum corresponding to thepolarized channel of the polar code with code length 2N and the unionbound calculation formula;

a module to determine the error probability threshold value 304,configured for determining error probability threshold values of thepolarized channels based on the union bounds of the polarized channelsand a predetermined measurement method;

a module to construct a polar code 305, configured for, given the codelength 2N and code rate K/2N, sorting error probability threshold valuesof all the polarized channels in ascending order, and selecting thepolarized channels corresponding to the K smallest error probabilitythreshold values to transmit information bits, while selecting the restof the polarized channels to transmit frozen bits, where K denotes thelength of information bits.

In the apparatus for constructing a polar code according to theembodiment of the present application: the polar weight spectrumcorresponding to each polarized channel of the polar code with codelength 2N is calculated, based on the polar weight spectrumcorresponding to each polarized channel of the polar code with codelength N and the MacWilliams identities; the union bound of eachpolarized channel under the condition of the additive white Gaussiannoise channel is calculated based on the polar weight spectrumcorresponding to each polarized channel of the polar code with codelength 2N and the union bound calculation formula; the error probabilitythreshold value of each polarized channel is determined based on theunion bound and a predetermined measurement method; given the codelength 2N and code rate K/2N, the error probability threshold values aresorted in ascending order, and the polarized channels corresponding tothe K smallest error probability threshold values are selected totransmit information bits, while the rest of the polarized channels areselected to transmit frozen bits. The construction method of polar codesprovided by the embodiments of this application can be SNR-independent,which reduces the complexity for constructing a polar code, and has agood practical prospect for the polar coded transmission system.

In an implementation of the present application, the module to determinethe error probability threshold value comprises:

a first sub-module, configured for, for each of the polarized channels,determining the union bound of the polarized channel as the errorprobability threshold value of the polarized channel; or

a second sub-module, configured for, for each of the polarized channels,determining a logarithmic form of the union bound of the polarizedchannel as the error probability threshold value of the polarizedchannel; or

a third sub-module, configured for, for each of the polarized channels,selecting an item corresponding to a minimum Hamming weight in thelogarithmic form of the union bound of the polarized channel as theerror probability threshold value of the polarized channel.

In an implementation of the present application, the second module tocalculate the polar weight spectrum is specifically configured for:calculating a weight spectrum S_(2N) ^((l))(p) corresponding to thel^(th) polarized channel of the polar code with code length 2N by usinga formula

${{S_{2N}^{(l)}(p)} = {\sum\limits_{h = l}^{2N}{A_{2N}^{(h)}(p)}}};$

calculating a weight spectrum S_(2N) ^((m))(p) corresponding to them^(th) polarized channel of the polar code with code length 2N based onthe MacWilliams identities

${{\sum\limits_{p = 0}^{{2N} - v}{\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}{S_{2N}^{(m)}(p)}}} = {2^{{2N} - m + 1 - v}{\sum\limits_{p = 0}^{v}{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix}{S_{2N}^{({{2N} + 2 - m})}(p)}}}}},{{{where}\mspace{14mu}\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}} = \frac{\left( {{2N} - p} \right)!}{{v!}{\left( {{2N} - p - v} \right)!}}},{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix} = \frac{\left( {{2N} - p} \right)!}{{\left( {{2N} - v} \right)!}{\left( {v - p} \right)!}}},{{{and}\mspace{14mu} v} = 0},1,\ldots\mspace{14mu},{{2N};}$

calculating the polar weight spectrum A_(2N) ^((m))(p) corresponding tothe m^(th) polarized channel of the polar code with code length 2N byusing A_(2N) ^((m))(p)=S_(2N) ^((m))(p)−S_(2N) ^((m+1))(p).

In an implementation of the present application, the module to determinethe union bound is specifically configured for: determining a unionbound of a q^(th) polarized channel under the condition of the additivewhite Gaussian noise channel as

${P_{UB}\left( W_{2N}^{(q)} \right)} = {\sum\limits_{p}^{\;}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}$by using a formula

${{P\left( W_{2N}^{(q)} \right)} \leq {\sum\limits_{p}^{\;}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}},$where q=1, 2, . . . , 2N, denotes an error probability of the q^(th)polarized channel, E_(s) denotes an average energy of the transmittedsignal, N₀ denotes noise power spectrum density, E_(s)/N₀ denotes asymbol signal-to-noise ratio (SNR).

An embodiment of the present application further provides an electronicdevice. Referring to FIG. 4, FIG. 4 is a structural diagram of anelectronic device according to an embodiment of the present application.The electronic device comprises: a processor 401, a communicationinterface 402, a memory 403 and a communication bus 404, where theprocessor 401, the communication interface 402 and the memory 403communicate with each other via the communication bus 404;

the memory 403 is configured for storing a computer program;

the processor 401 is configured for implementing steps of any method forconstructing a polar code when executing the computer program stored inthe memory 403.

It should be understood that, the communication bus 404 mentioned forthe electronic device may be a Peripheral Component Interconnect (PCI)bus or an Extended Industry Standard Architecture (EISA) bus and thelike. The communication bus 404 can be classified as an address bus, adata bus, a control bus and the like. In FIG. 4, only a thick line isused, but it does not mean that there is only one bus or one type ofbus.

The communication interface 402 is configured for communication betweenthe above electronic device and other devices.

The memory 403 may comprise a Random Access Memory (RAM), may furthercomprise a non-volatile memory, for example, at least one disk memory.Optionally, the memory may also be at least one storage device locatedfar from the processor.

The above processor 401 may be a general-purpose processor, comprising:a Central Processing Unit (CPU), a Network Processor (NP) and the like;or a Digital Signal Processing (DSP), an Application Specific IntegratedCircuit (ASIC), a Field-Programmable Gate Array (FPGA) or otherprogrammable logic devices, discrete gates or transistor logic devices,discrete hardware components.

In the electronic device of the present application, by executing theprogram stored in the memory, the processor: calculates the polar weightspectrum corresponding to each polarized channel of the polar code withcode length 2N, based on polar weight spectrums corresponding to eachpolarized channel of the polar code with code length N and theMacWilliams identities; calculates a union bound of each of thepolarized channels under the condition of the additive white Gaussiannoise channel based on the polar weight spectrum corresponding to eachpolarized channel in the polar code with code length 2N and a unionbound calculation formula; determines an error probability thresholdvalue of each polarized channel based on the union bound and apredetermined measurement method; given the code length 2N and code rateK/2N, sorts error probability threshold values in ascending order,selects polarized channels corresponding to the K smallest errorprobability threshold values to transmit information bits, and selectsthe rest of the polarized channels to transmit frozen bits. Theconstruction method of polar codes provided by the embodiments of thisapplication can be SNR-independent, which reduces the complexity forconstructing a polar code, and has a good practical prospect for thepolar coded transmission system.

An embodiment of the present application further provides acomputer-readable storage medium. The computer-readable medium stores acomputer program therein, when being executed by a processor, implementssteps of any method for constructing a polar code.

The instructions, when being executed by a computer, stored in thecomputer-readable storage medium according to the embodiment of thepresent application: calculate the polar weight spectrum correspondingto each polarized channel of the polar code with code length 2N, basedon polar weight spectrums corresponding to each polarized channel of thepolar code with code length N and the MacWilliams identities; calculatea union bound of each of the polarized channels under the condition ofthe additive white Gaussian noise channel based on the polar weightspectrum corresponding to each polarized channel in the polar code withcode length 2N and a union bound calculation formula; determine an errorprobability threshold value of each polarized channel based on the unionbound and a predetermined measurement method; given the code length 2Nand code rate K/2N, sort error probability threshold values in ascendingorder, select polarized channels corresponding to the K smallest errorprobability threshold values to transmit information bits, and selectthe rest of the polarized channels to transmit frozen bits. Theconstruction method of polar codes provided by the embodiments of thisapplication can be SNR-independent, which reduces the complexity forconstructing a polar code, and has a good practical prospect for thepolar coded transmission system.

The embodiments described above are simply preferable embodiments of thepresent application, and are not intended to limit the scope ofprotection of the present application. Any modifications, alternatives,improvements, or the like within the spirit and principle of the presentapplication shall be included within the scope of protection of thepresent application.

The invention claimed is:
 1. A method for constructing a polar code,comprising: calculating a polar weight spectrum A_(2N) ^((l))(p)corresponding to an l^(th) polarized channel of a polar code with codelength 2N based on a polar weight spectrum A_(N) ^((i))(j) correspondingto an i^(th) polarized channel of a polar code with code length N and aformula $\left\{ {\begin{matrix}{{{A_{2N}^{(l)}(p)} = {A_{N}^{(i)}(j)}},{p = {2j}},} \\{{{A_{2N}^{(l)}(p)} = 0},{p\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{positive}\mspace{14mu}{odd}\mspace{14mu}{number}\mspace{14mu}{less}\mspace{14mu}{then}\mspace{14mu} 2N}}\end{matrix},} \right.$ where i=1, 2, . . . , N,l=i+N, j=0, 1, 2, . . .,N, j denotes a weight of a codeword in the polar code with code lengthN, p denotes a weight of a codeword in the polar code with code length2N, N=2^(x), x is a positive integer; calculating a polar weightspectrum A_(2N) ^((m))(p) corresponding to an m^(th) polarized channelof the polar code with code length 2N, based on the polar weightspectrum A_(2N) ^((l))(p) corresponding to the l^(th) polarized channeland MacWilliams identities, where m=1, 2, . . . , N; calculating, foreach polarized channel of the polar code with code length 2N, a unionbound on an error probability of a polarized channel under a conditionof additive white Gaussian noise channel based on the polar weightspectrum corresponding to the polarized channel and a union boundcalculation formula; determining error probability threshold values ofthe polarized channels based on the union bounds of the polarizedchannels and a predetermined measurement method; given the code length2N and code rate K/2N, sorting the error probability threshold values ofall the polarized channels in ascending order, and selecting polarizedchannels corresponding to K smallest error probability threshold valuesto transmit information bits, while selecting remaining polarizedchannels to transmit frozen bits; where K denotes a length ofinformation bits.
 2. The method for constructing a polar code of claim1, wherein, determining the error probability threshold values of thepolarized channels based on the union bounds of the polarized channelsand a predetermined measurement method, comprises: for each of thepolarized channels, determining the union bound of the polarized channelas the error probability threshold value of the polarized channel; ordetermining a logarithmic form of the union bound of the polarizedchannel as the error probability threshold value of the polarizedchannel; or selecting an item corresponding to a minimum Hamming weightin the logarithmic form of the union bound of the polarized channel asthe error probability threshold value of the polarized channel.
 3. Themethod for constructing a polar code of claim 1, wherein, calculatingthe polar weight spectrum A_(2N) ^((m))(p) corresponding to the m^(th)polarized channel of the polar code with code length 2N, based on thepolar weight spectrum A_(2N) ^((l))(p) corresponding to the l^(th)polarized channel and the MacWilliams identities, comprises: calculatingthe weight spectrum S_(2N) ^((l))(p) corresponding to the l^(th)polarized channel of the polar code with code length 2N based on aformula${{S_{2N}^{(l)}(p)} = {\sum\limits_{h = l}^{2N}{A_{2N}^{(h)}(p)}}};$calculating the weight spectrum S_(2N) ^((m))(p) corresponding to them^(th) polarized channel of the polar code with code length 2N based onthe MacWilliams identities${{\sum\limits_{p = 0}^{{2N} - v}{\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}{S_{2N}^{(m)}(p)}}} = {2^{{2N} - m + 1 - v}{\sum\limits_{p = 0}^{v}{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix}{S_{2N}^{({{2N} + 2 - m})}(p)}}}}},{{{where}\mspace{14mu}\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}} = \frac{\left( {{2N} - p} \right)!}{{v!}{\left( {{2N} - p - v} \right)!}}},{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix} = \frac{\left( {{2N} - p} \right)!}{{\left( {{2N} - v} \right)!}{\left( {v - p} \right)!}}},{{{and}\mspace{14mu} v} = 0},1,\ldots\mspace{14mu},{{2N};}$calculating the polar weight spectrum A_(2N) ^((m))(p) corresponding tothe m^(th) polarized channel of the polar code with code length 2N basedon A_(2N) ^((m))(p)=S_(2N) ^((m))(p)−S_(2N) ^((m+1))(p).
 4. The methodfor constructing a polar code of claim 1, wherein, calculating, for eachpolarized channel, a union bound on an error probability of thepolarized channel under the condition of additive white Gaussian noisechannel based on the polar weight spectrum corresponding to thepolarized channel of the polar code with code length 2N and the unionbound calculation formula, comprises: determining a union bound of aq^(th) polarized channel under the condition of the additive whiteGaussian noise channel as${P_{UB}\left( W_{2N}^{(q)} \right)} = {\sum\limits_{p}^{\;}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}$by using a formula${{P\left( W_{2N}^{(q)} \right)} \leq {\sum\limits_{p}^{\;}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}},$where q=1, 2, . . . , 2N, P(W_(2N) ^((q))) (denotes an error probabilityof the q^(th) polarized channel, E_(s), denotes an average energy of atransmitted signal, N₀ denotes noise power spectrum density, E_(s)/N₀denotes a symbol signal-to-noise ratio (SNR).
 5. An apparatus forconstructing a polar code, comprising: a first module to calculate apolar weight spectrum, configured for calculating a polar weightspectrum A_(2N) ^((l))(p) corresponding to an l^(th) polarized channelof a polar code with code length 2N based on a polar weight spectrumA_(N) ^((i))(j) corresponding to an i^(th) polarized channel of a polarcode with code length N and a formula $\left\{ {\begin{matrix}{{{A_{2N}^{(l)}(p)} = {A_{N}^{(i)}(j)}},{p = {2j}},} \\{{{A_{2N}^{(l)}(p)} = 0},{p\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{positive}\mspace{14mu}{odd}\mspace{14mu}{number}\mspace{14mu}{less}\mspace{14mu}{then}\mspace{14mu} 2N}}\end{matrix},} \right.$ where i=1, 2, . . . , N, l=i+N, j=0, 1, 2, . . ., N, j denotes a weight of a codeword in the polar code with code lengthN, p denotes a weight of a codeword in the polar code with code length2N, N=2^(x),x is a positive integer; a second module to calculate apolar weight spectrum, configured for calculating a polar weightspectrum A_(2N) ^((m))(p) corresponding to an m^(th) polarized channelof the polar code with code length 2N, based on the polar weightspectrum A_(2N) ^((l))(p) corresponding to the l^(th) polarized channeland MacWilliams identities, where m=1, 2, . . . , N; a module todetermine a union bound, configured for calculating, for each polarizedchannel of the polar code with code length 2N, a union bound on an errorprobability of a polarized channel under a condition of additive whiteGaussian noise channel based on the polar weight spectrum correspondingto the polarized channel and a union bound calculation formula; a moduleto determine error probability threshold values, configured fordetermining error probability threshold values of the polarized channelsbased on the union bounds of the polarized channels and a predeterminedmeasurement method; a module to construct a polar code, configured for,given the code length 2N and code rate K/2N, sorting error probabilitythreshold values of all the polarized channels in ascending order, andselecting the polarized channels corresponding to K smallest errorprobability threshold values to transmit information bits, whileselecting remaining polarized channels to transmit frozen bits, where Kdenotes the length of information bits.
 6. The apparatus forconstructing a polar code of claim 5, wherein, the module to determinethe error probability threshold value comprises: a first sub-module,configured for, for each of the polarized channels, determining theunion bound of the polarized channel as the error probability thresholdvalue of the polarized channel; or a second sub-module, configured for,for each of the polarized channels, determining a logarithmic form ofthe union bound of the polarized channel as the error probabilitythreshold value of the polarized channel; or a third sub-module,configured for, for each of the polarized channels, selecting an itemcorresponding to a minimum Hamming weight in the logarithmic form of theunion bound of the polarized channel as the error probability thresholdvalue of the polarized channel.
 7. The apparatus for constructing apolar code of claim 5, wherein, the second module to calculate a polarweight spectrum is specifically configured for: calculating a weightspectrum S_(2N) ^((l))(p) corresponding to the l^(th) polarized channelof the polar code with code length 2N by using a formula${{S_{2N}^{(l)}(p)} = {\sum\limits_{h = l}^{2N}{A_{2N}^{(h)}(p)}}};$calculating a weight spectrum S_(2N) ^((m)) (p) corresponding to them^(th) polarized channel of the polar code with code length 2N based onthe MacWilliams identities${{\sum\limits_{p = 0}^{{2N} - v}{\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}{S_{2N}^{(m)}(p)}}} = {2^{{2N} - m + 1 - v}{\sum\limits_{p = 0}^{v}{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix}{S_{2N}^{({{2N} + 2 - m})}(p)}}}}},{{{where}\mspace{14mu}\begin{pmatrix}{{2N} - p} \\v\end{pmatrix}} = \frac{\left( {{2N} - p} \right)!}{{v!}{\left( {{2N} - p - v} \right)!}}},{\begin{pmatrix}{{2N} - p} \\{{2N} - v}\end{pmatrix} = \frac{\left( {{2N} - p} \right)!}{{\left( {{2N} - v} \right)!}{\left( {v - p} \right)!}}},{{{and}\mspace{14mu} v} = 0},1,\ldots\mspace{14mu},{{2N};}$calculating the polar weight spectrum A_(2N) ^((m))(p) corresponding tothe m^(th) polarized channel of the polar code with code length 2N byusing A_(2N) ^((m))(p)=S_(2N) ^((m))(p)−S_(2N) ^((m+l))(p).
 8. Theapparatus for constructing a polar code of claim 5, wherein, the moduleto determine a union bound is specifically configured for: determining aunion bound of a q^(th) polarized channel under the condition of theadditive white Gaussian noise channel as${{P\left( W_{2N}^{(q)} \right)} \leq {\sum\limits_{p}^{\;}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}},$by using a formula${P_{UB}\left( W_{2N}^{(q)} \right)} = {\sum\limits_{p}^{\;}{{A_{2N}^{(q)}(p)}\exp\left\{ {{- p}\frac{E_{s}}{N_{0}}} \right\}}}$where q =1, 2, . . . , 2N, P(W_(2N) ^((q))) denotes an error probabilityof the q^(th) polarized channel, E_(s), denotes an average energy of atransmitted signal, N₀ denotes noise power spectrum density, E_(s)/N₀denotes a symbol signal-to-noise ratio (SNR).
 9. An electronic device,comprising: a processor, a communication interface, a memory and acommunication bus, where the processor, the communication interface andthe memory communicate with each other via the communication bus; thememory is configured for storing a computer program; the processor isconfigured for implementing steps of the method for constructing a polarcode of claim 1 when executing the computer program stored in thememory.
 10. A non-transitory computer-readable storage medium, wherein,the computer-readable storage medium stores a computer program therein,and the computer program, when being executed by a processor, implementssteps of the method for constructing a polar code of claim 1.